Properties

Label 2-31200-1.1-c1-0-47
Degree $2$
Conductor $31200$
Sign $-1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·7-s + 9-s + 5·11-s − 13-s + 5·17-s + 4·19-s − 3·21-s − 2·23-s + 27-s − 9·29-s + 3·31-s + 5·33-s + 10·37-s − 39-s − 12·41-s − 2·43-s − 9·47-s + 2·49-s + 5·51-s − 9·53-s + 4·57-s + 3·59-s − 7·61-s − 3·63-s − 9·67-s − 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 1.21·17-s + 0.917·19-s − 0.654·21-s − 0.417·23-s + 0.192·27-s − 1.67·29-s + 0.538·31-s + 0.870·33-s + 1.64·37-s − 0.160·39-s − 1.87·41-s − 0.304·43-s − 1.31·47-s + 2/7·49-s + 0.700·51-s − 1.23·53-s + 0.529·57-s + 0.390·59-s − 0.896·61-s − 0.377·63-s − 1.09·67-s − 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.22249936687952, −14.80070942549435, −14.28256728630336, −13.87994049858174, −13.17893736499907, −12.84266938475838, −12.14665442539698, −11.70543934418052, −11.27686124876051, −10.24146191280443, −9.817017057713700, −9.478940189565012, −9.081601278371063, −8.238992927871516, −7.714230300475693, −7.147049974307984, −6.478777654316654, −6.099455680461284, −5.331172631154903, −4.551942492247865, −3.735290119210190, −3.388000353833976, −2.851460143436357, −1.754989649706163, −1.183036604268919, 0, 1.183036604268919, 1.754989649706163, 2.851460143436357, 3.388000353833976, 3.735290119210190, 4.551942492247865, 5.331172631154903, 6.099455680461284, 6.478777654316654, 7.147049974307984, 7.714230300475693, 8.238992927871516, 9.081601278371063, 9.478940189565012, 9.817017057713700, 10.24146191280443, 11.27686124876051, 11.70543934418052, 12.14665442539698, 12.84266938475838, 13.17893736499907, 13.87994049858174, 14.28256728630336, 14.80070942549435, 15.22249936687952

Graph of the $Z$-function along the critical line